eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-01
11:1
11:13
10.4230/LIPIcs.ESA.2022.11
article
Bounding and Computing Obstacle Numbers of Graphs
Balko, Martin
1
https://orcid.org/0000-0001-9688-9489
Chaplick, Steven
2
https://orcid.org/0000-0003-3501-4608
Ganian, Robert
3
https://orcid.org/0000-0002-7762-8045
Gupta, Siddharth
4
https://orcid.org/0000-0003-4671-9822
Hoffmann, Michael
5
https://orcid.org/0000-0001-5307-7106
Valtr, Pavel
1
https://orcid.org/0000-0002-3102-4166
Wolff, Alexander
6
https://orcid.org/0000-0001-5872-718X
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Maastricht University, The Netherlands
Algorithms and Complexity Group, TU Wien, Austria
Department of Computer Science, University of Warwick, Coventry, UK
Department of Computer Science, ETH Zürich, Switzerland
Institut für Informatik, Universität Würzburg, Germany
An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons.
It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)²) [Dujmović and Morin, 2015]. We improve this lower bound to Ω(n/log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n²) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances.
We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol244-esa2022/LIPIcs.ESA.2022.11/LIPIcs.ESA.2022.11.pdf
Obstacle representation
Obstacle number
Visibility
NP-hardness
FPT